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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
Our full course list for upcoming classes is below:
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0 replies
jlacosta
Apr 2, 2025
0 replies
Short combi omg
Davdav1232   5
N 3 minutes ago by fagot
Source: Israel TST 2025 test 4 p3
Let \( n \) be a positive integer. A graph on \( 2n - 1 \) vertices is given such that the size of the largest clique in the graph is \( n \). Prove that there exists a vertex that is present in every clique of size \( n\)
5 replies
Davdav1232
Feb 3, 2025
fagot
3 minutes ago
Isi 2016 geometry
zizou10   22
N 6 minutes ago by kamatadu
Source: ISI BSTAT 2016 #5
Prove that there exists a right angle triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in Arithmetic Progression

Here $d$ is an integer.
22 replies
1 viewing
zizou10
May 8, 2016
kamatadu
6 minutes ago
If ab+1 is divisible by A then so is a+b
ravengsd   3
N 10 minutes ago by trigadd123
Source: Romania EGMO TST 2025 Day 2, Problem 4
Find the greatest positive integer $A$ such that, for all positive integers $a$ and $b$, if $A$ divides $ab+1$, then $A$ divides $a+b$.
3 replies
ravengsd
4 hours ago
trigadd123
10 minutes ago
IMO Shortlist 2012, Geometry 2
lyukhson   88
N 20 minutes ago by zuat.e
Source: IMO Shortlist 2012, Geometry 2
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.
88 replies
lyukhson
Jul 29, 2013
zuat.e
20 minutes ago
Putnam 1958 February A5
sqrtX   4
N 4 hours ago by Safal
Source: Putnam 1958 February
Show that the integral equation
$$f(x,y) = 1 + \int_{0}^{x} \int_{0}^{y} f(u,v) \, du \, dv$$has at most one solution continuous for $0\leq x \leq 1, 0\leq y \leq 1.$
4 replies
sqrtX
Jul 18, 2022
Safal
4 hours ago
Miklós Schweitzer 1956- Problem 1
Coulbert   1
N 5 hours ago by NODIRKHON_UZ
1. Solve without use of determinants the following system of linear equations:

$\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k$ ($k= 0,1, \dots , n$),

where $\alpha$ is a fixed real number. (A. 7)
1 reply
Coulbert
Oct 9, 2015
NODIRKHON_UZ
5 hours ago
D1021 : Does this series converge?
Dattier   3
N 5 hours ago by Dattier
Source: les dattes à Dattier
Is this series $\sum \limits_{k\geq 1} \dfrac{\ln(1+\sin(k))} k$ converge?
3 replies
Dattier
Apr 26, 2025
Dattier
5 hours ago
If a matrix exponential is identity, does it follow the initial matrix is zero?
bakkune   5
N 6 hours ago by loup blanc
This might be a really dumb question, but I have neither a rigorous proof nor a counter example.

For any square matrix $\mathbf{A}$, define
$$
e^{\mathbf{A}} = \mathbf{I} + \sum_{n=1}^{+\infty} \frac{1}{n!}\mathbf{A}^n
$$where $\mathbf{I}$ is the identity matrix. If for some matrix $\mathbf{A}$ that $e^{\mathbf{A}}$ is identity, does it follow that $\mathbf{A}$ is zero?
5 replies
bakkune
Mar 4, 2025
loup blanc
6 hours ago
Range of 2 parameters and Convergency of Improper Integral
Kunihiko_Chikaya   3
N Today at 11:37 AM by Mathzeus1024
Source: 2012 Kyoto University Master Course in Mathematics
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.
3 replies
Kunihiko_Chikaya
Aug 21, 2012
Mathzeus1024
Today at 11:37 AM
Matrix Row and column relation.
Schro   6
N Today at 6:20 AM by Schro
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
6 replies
Schro
Apr 28, 2025
Schro
Today at 6:20 AM
A small problem in group theory
qingshushuxue   2
N Today at 4:42 AM by qingshushuxue
Assume that $G,A,B,C$ are group. If $G=\left( AB \right) \bigcup \left( CA \right)$, prove that $G=AB$ or $G=CA$.

where $$A,B,C\subset G,AB\triangleq \left\{ ab:a\in A,b\in B \right\}.$$
2 replies
qingshushuxue
Today at 2:06 AM
qingshushuxue
Today at 4:42 AM
Putnam 1958 February A4
sqrtX   2
N Today at 2:14 AM by centslordm
Source: Putnam 1958 February
If $a_1 ,a_2 ,\ldots, a_n$ are complex numbers such that
$$ |a_1| =|a_2 | =\cdots = |a_n| =r \ne 0,$$and if $T_s$ denotes the sum of all products of these $n$ numbers taken $s$ at a time, prove that
$$ \left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n}$$whenever the denominator of the left-hand side is different from $0$.
2 replies
sqrtX
Jul 18, 2022
centslordm
Today at 2:14 AM
analysis
Hello_Kitty   2
N Yesterday at 10:37 PM by Hello_Kitty
what is the range of $f=x+2y+3z$ for any positive reals satifying $z+2y+3x<1$ ?
2 replies
Hello_Kitty
Yesterday at 9:59 PM
Hello_Kitty
Yesterday at 10:37 PM
Putnam 1958 February A1
sqrtX   2
N Yesterday at 10:32 PM by centslordm
Source: Putnam 1958 February
If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying
$$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.
2 replies
sqrtX
Jul 18, 2022
centslordm
Yesterday at 10:32 PM
n variables with n-gon sides
mihaig   0
Apr 25, 2025
Source: Own
Let $n\geq3$ and let $a_1,a_2,\ldots, a_n\geq0$ be reals such that $\sum_{i=1}^{n}{\frac{1}{2a_i+n-2}}=1.$
Prove
$$\frac{24}{(n-1)(n-2)}\cdot\sum_{1\leq i<j<k\leq n}{a_ia_ja_k}\geq3\sum_{i=1}^{n}{a_i}+n.$$
0 replies
mihaig
Apr 25, 2025
0 replies
n variables with n-gon sides
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mihaig
7352 posts
#1
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Let $n\geq3$ and let $a_1,a_2,\ldots, a_n\geq0$ be reals such that $\sum_{i=1}^{n}{\frac{1}{2a_i+n-2}}=1.$
Prove
$$\frac{24}{(n-1)(n-2)}\cdot\sum_{1\leq i<j<k\leq n}{a_ia_ja_k}\geq3\sum_{i=1}^{n}{a_i}+n.$$
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